The term interdependency is defined as: “A bidirectional relationship between two infrastructures through which the state of each infrastructure influences or is correlated to the state of the other” by Rinaldi et al. (2001). Modeling the lifelines as a system of networks with proper dependency considerations instead of treating them as independent networks is one approach towards more accurate anticipation of the effects of earthquakes (Kim, 2007). A system of systems approach requires modeling
multiple but individually operational networks that would interact with each other to satisfy specific demands (Satumtira and Duenas-Osorio, 2010).
Interdependencies among lifeline networks can be described via numerous occurrences. Lifeline networks of San Francisco experienced extensive damage due to the Loma Prieta Earthquake in 1989. Water supply system on the lower zones of the city had failed because of damaged pipes and hydrants. A system wide disruption had been witnessed on the water network due to cascading failures, which lead to the loss of firefighting abilities. Also, an exploding gas transmission line in New York in 1989 that caused power loss for about 5 hours can be given as another example of lifeline interdependency (O'Rourke, 1994). Infrastructure interdependencies can be defined as failures within a system due to an initial infrastructural failure on a different system caused by an extreme event (McDaniels et al., 2008). Modeling of network interdependencies is a highly complex task, given the different nature of each physical or spatial connection between systems. Depending on the extent of the study, several disciplines such as civil engineering, mechanical engineering, electrical engineering, computer science, economics and other social sciences may have to get involved. Failure to identify the complex behavior of interacting systems and the underlying dynamics of the complex structure of interdependent systems (Figure 2.2) would result in inadequate response and poor performance in the prepared plans which may cause loss of public trust and human life (Pederson et al., 2006).
Heller (2002) stated the complexity of system behaviors with increasing automation in the infrastructure systems and pointed out the possibility that system interdependencies may cause global perturbations more severe than previously expected. Integration of information systems in infrastructure operations targets the achievement of new technologies, overcoming limitations, and increasing operational efficiency. Focus on infrastructure interdependencies was initiated following the Presidential Decision Directive Number 63 (Clinton, 1998) which revealed the rapidly growing exploitation potential of energy, banking, finance, transportation, lifelines, and telecommunications systems. Incapacity or destruction of these systems were said to cause severe impacts on United States’ defense and economic security. Heller (2002) also described infrastructure systems as complex adaptive systems
nonlinear, and spatiotemporal interactions among sub-systems or components. Modeling of complex systems is a major challenge for the researchers because of the unpredictable outcomes of the complicated interactions. Complex adaptive systems are used to harness this complexity of these interactions (Axelrod and Cohen, 1999).
Figure 2.2 : Infrastructure interdepenedencies (Pederson et al., 2006).
Rinaldi et al. (2001) presented a conceptual framework for identification, definition, and modeling of critical infrastructure interdependencies. According to the framework, there exist six dimensions of infrastructure interdependencies intending to define, understand, and model the interdependencies. The six dimensions of infrastructure interdependencies are: type of interdependency, coupling and response behavior, failure type, infrastructure characteristics, and state of operations (Figure 2.3).
Figure 2.3 : Dimensions of infrastructure interdependencies (Rinaldi et al., 2001). System interactions may vary depending on the interdependency type where physical interdependency suggests the state of each infrastructure is influenced by the material output of another; cyber interdependency suggests the state of an infrastructure is influenced by the information transmitted through the information system; geographic interdependency suggests a local event altering the states of the infrastructure systems in proximity such as an explosion causing correlated disturbances; and logical interdependency suggests the state of an infrastructure system is dependent of the state of another via a non-physical, non-geographical, and non-cyber connection (Rinaldi et al., 2001).
Analyzing infrastructure systems with a system-of-systems perspective with interdependency considerations would lead to enhanced validity of analyses and better, more appropriate policies and decision regarding emergencies with severe disruptions on infrastructure systems. However, each existing modeling and simulation methodology addresses one or more factors associated with interdependencies that complicate analysis efforts (Table 2.2). It is suggested that multidisciplinary approaches may result in the development of an all-encompassing methodology (Rinaldi, 2004).
Table 2.2 : Factors affecting interdependency analyses (Rinaldi, 2004). Factor Implication for Analyses
Time Scales
Different infrastructures have varying time scales of importance, varying from milliseconds to years.
Geographic Scales
Issues range from cities to national or international levels in scale. Scale affects the resolution and quantity of infrastructure and interdependency data required for models.
Cascading Effects
Disruptions in one infrastructure can ripple or cascade into other infrastructures, creating second and higher order disruptions.
Social/Psychological Elements
Social networks and behavioral responses can influence infrastructure operations, such as the spread of an infectious disease and the response of the public health infrastructure.
Operational Procedures
Company-specific procedures influence the state of an infrastructure, such as responses to market fluctuations. Business Policies
Specific corporate business policies affect the operations of the infrastructures.
Restoration and Recovery Procedures
Company-specific procedures influence the state of an infrastructure during a crisis or emergency, and may affect coordination among various infrastructure owners. Cross-infrastructure restoration/recovery procedures may not exist.
Legal and
Regulatory Regimes
Government actions will influence operational behaviors as well as the response to and recovery from disasters or disruptions.
Stakeholder Concerns
Stakeholders have differing motivations and different sets of concerns that drive modeling and simulation requirements.
Given the complex behavior of interdependent systems, modeling and simulation efforts would provide approximate information on the consequences of rare extreme events, however would not be accurately representing them. Although the stakeholders of infrastructure systems have extensive experience regarding daily small-scale outages and disruptions, the limited experience against major infrastructure failures requires the utilization such modeling and simulation efforts which would provide valuable insight for development of mitigation, response and recovery plans. With the inadequacy of historical record and experience, simulation of interdependent system models offer the only guidance available (Rinaldi, 2004). Additionally, verification and validation of the existing models possess great importance for the development and improvement of existing methodologies. This
can be achieved by taking historical information or commonly acknowledged models as benchmarks for testing and calibrating new models.
In the presence of numerous uncertainties within the available models, it is necessary to obtain essential data for inventory validation in order to result in assessments as accurate as possible. Acquisition, updating, verification, and validation of data have fundamental importance in modeling and present one of the biggest obstacles in the process according to Rinaldi (2000). Since most infrastructure systems are owned and operated by private entities, crucial information for the development are not easily and directly accessible. Most data owners abstain from sharing information due to concerns about possible confidentiality, privacy, liability, security, and legal issues that may be faced.
Interdependency concept was first realized and investigated in economic and social models. Haavelmo (1943) stated the improbability to utilize a suitable method for statistical assessment of an economic relation without taking the set of relationships with the theoretical model which it is originating from into account. Such models which are characterized by mutual interactions between the variables are defined as interdependent models. Interdependency theory was developed by Thibaut and Kelley (1959) for social models with the aim to clarify the life-space representation of human motivation where all interdependencies were defined via four parameters: degree of dependency, mutuality of dependency, correspondence of outcomes, and the basis for dependency. Wilson and Pownall (1976) emphasized the aggregation difficulties of interacting systems via urban models. They proposed a micro-level interdependency model which would overcome the problems caused by the weak connections between the subsystems within an urban system. Boissevain (1979) pointed out the significance of interdependency by stating that interdependencies would provide a framework where separation of micro and macro analytical levels from each other was difficult. The focus on interdependency in network analysis also enables modeling of secondary agents resulting from the interactions. Gee and Treuner (1981) published one of the earliest works focusing on infrastructure interdependency by proposing a regional development planning model for a system of interdependent networks. Victor and Blackburn (1987) used the interdependence theory by Thibaut and Kelley to investigate inter-unit conflict and the outcomes of
criteria decision making analysis used to make the initial assumption of independent criteria. Carlsson and Fullér (1995) developed a systematic procedure to determine and analyze the interdependencies for the solutions of multi-criteria decision making problems.
Although interdependent models are widely applied in business and economics, considering the complex interacting structures of lifeline systems and computational capabilities, there also exists application opportunities in lifeline earthquake engineering studies. Interdependent models were utilized for estimation of economic impacts on regional systems of agriculture, mining, construction, etc. following a major disruption on power distribution systems caused by catastrophic earthquakes (Rose et al., 1997). The ASPEN-EE tool, developed by Barton et al. (2000) provides an agent-based model to assess possible power outages and their impacts on market structures performing decision-making processes via agent interactions. The interactions were modeled between household, commercial, industrial, governmental, generation company, system operator, fuel company, disaster, bulletin board, and weather agents.
Within the context of earthquake engineering, network interactions and interdependencies are taken into consideration to determine the system performance after a perturbation (Adachi and Ellingwood, 2008) or to consider the effects of restoration efforts to the network performance (Lee II et al., 2007; Shinozuka et al., 2005).
Duenas-Osorio (2005) developed a model composed of network systems with multiple levels of interdependencies based on spatial proximity. Instead of a macro- level approach to the interacting systems, the model focused on network topology (physical layout) and flow patterns. In this model, topology of a network is characterized by a number of parameters: Mean Distance (L) – measure of the shortest paths between each vertice pair; Vertex Degree (d(v)) – the number of edges at vertex v; Clustering Coefficient (γ) – characterizing the extent to which vertices adjacent to a vertex v are adjacent to each other; and Redundancy Ratio (RR) - a measure of the number of different paths from a vertex to each of the vertices within the set of the neighbors of its neighbors. Duenas-Osorio (2005) also defined three performance measures for functionality characterization of a network: Efficiency, connectivity loss and service flow reduction. These measures assess the network
performance with metrics depending on the topological settings of the network, or with more detailed metrics depending on supply, demand, and flow patterns additional to the topological settings. Connectivity Loss (CL) measures the ability of
every distribution node to receive flow from generation nodes. Service Flow Reduction (SFR) determines the amount of flow that the system can provide based on
the demand before the disturbance.
Kim (2007) has proposed a methodology based on the network structure defined by Duenas-Osorio (2005) with further clarification on the probabilistic interdependency model, modified failure models and improved interdependent failure mechanisms. The model is formulated over electric power and water network systems with water system being dependent on electric power, based on the fact that electricity is vital for the operation of almost every function in urban societies (Shinozuka et al., 2005). CL and SFR measures were utilized for the interacting networks in order to quantify
the functional loss of a system when some of the components are likely to be dysfunctional. Each network is built of links and nodes; with links representing power lines or water pipes, nodes representing network facility structures. Nodes are classified as generation, intermediate, or distribution in each network where flow in the network is generated by generation nodes, and is discharged by distribution nodes. The interactions were defined between the networks where water generation nodes are dependent to electric power supplied to the system by power distribution nodes. The failure of a component after an earthquake is linked to two main reasons in the model: Failure due to earthquake damage, and non-functionality of a network component due to power outage. Power outage can be caused by earthquake damage to the distribution facility, or failure of the nodes and links in the power network feeding electric power to the distribution node. Furthermore, although being functional and not affected by interdependency, a network node can still fail by losing its connectivity to the network. This happens when a generation node has no surviving outgoing links, or when a distribution node has no surviving incoming links, thus being isolated from the network (Kim, 2007). The effects of network interconnection on post-seismic serviceability are presented in Figure 2.4 and Figure 2.5.
Figure 2.4 : Effect of interdependency on connectivity loss (Kim, 2007).
Satumtira and Duenas-Osorio (2010) classified interdependent modeling based on existing literature in a hierarchial manner under six main categories: mathematical model, modeling objective, analysis scale, quantity and quality of input data, targeted discipline, and end-user type (Figure 2.6). It was stressed out that most of the existing interdependent modeling strategies were complimentary to each other rather than competing and hybrid modeling techniques that would be the combination of multiple approaches would be beneficial for the future of lifelines interdependency modeling. The study also specify that while early models developed for lifelines interdependency were taking agent-based approaches for simulation of specialized and diverse tasks within lifeline systems, most recent models are mostly probabilistic models based on graph and network theories. Probabilistic network models provide more accurate visual representation of the physical structure of lifeline systems with a less data-intensive approach than agent-based models. Utilization of network and graph theories enables modeling of various types of infrastructure components and their coupling topology with accurate visual representation. In such models, interdependencies are defined with adjacency matrices specifying the agents, direction, and the strength of couplings via conditional probabilities (Satumtira and Duenas-Osorio, 2010).
According to the hierarchical classification of interdependent models, the methodology that is utilized in this study can be specified as adopting a network model based on graph theory as its mathematical model; with topological structuring of the analyzed networks and utilizing interdependency tables to define the interactions between network agents. The objective of the model is to aid risk and vulnerability assessment studies by providing estimations on seismic loss and system serviceability on a system of systems scale. The input data for the analyses is required to be gathered from the data owning infrastructural service providers through mutual agreements. The main target discipline of the methodology is engineering. However since the analysis output may be used to assess the affected population, it is possible to utilize the analysis methodology on social models. Finally, the end users to benefit from the analyses would be academic researchers who want to develop more accurate and improved models or governmental agencies who would use the assessment results on disaster management planning efforts.
Figure 2.6 : Hierarchial classification of interdependent models (Satumtira and Duenas-Osorio, 2010).